How does the work of CNC mills compare to that of a master wood carver? I want to make some furniture with carved wooden legs. The cost of commissioning a master carver to do this work would be pretty astronomical, however, so I am wondering if I could get just a good result by using CNC.

You are paying for expertise in either case, or DIY to substitute sweat equity for that expertise cost.
– jdvMar 18 at 18:52

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If you want to make 4 identical legs, the CNC machine is the way to go. It will do exactly what you ask of it thousands of times. That said, it's takes some skill to design the turning in CAD to get the design you actually want. If you want to your legs to look hand carved, then hand carve them - the minor variations of a master will make it look authentic, and the major variations of an amateur will make it look... like you're practicing. :)
– FreeManMar 18 at 19:18

2 Answers
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To do these chair legs you would need a large mill with rotary table to carve the legs on all sides $$$$.

The mill uses a circular cutter which will leave rounded corners.

Freeman earlier said it would take some skill to do the 3D design. To see just how difficult install Fusion 360, one of the most popular programs to handle something of this complexity. Free for hobbyists. Try a really simple flat carving about the size of a pack of cards and get it 3D printed. The same design can be sent to a CNC mill or a 3D printer. Same code, just one adds material the other removes material.

CNC machines can carve impressive carvings but depend on the programing, most can not carve true 3 dimensional (statues etc.) and are lacking, but repeatable, also I can not carve. Often hand carving is faster, has more feeling, depth, ... and always more artistic.